Thus, we have no notions of “bit sequence”,“code”, “coding”, “decoding” at all. But their ghosts still haunt us: the definition of norm function looks rather strange without thinking on the fact that is was transferred from a concept of coding.

Thus, we have no notions of “bit sequence”,“code”, “coding”, “decoding” at all. But their ghosts still haunt us: the definition of norm function looks rather strange without thinking on the fact that is was transferred from a concept of coding.

2.1 Coding

of course, c, d are meta-variables, and also some other notations are changed slightly.

2.2 Decoding

Having seen this, decoding is rather straightforward.
Here is a parser for illustration, but it serves only didactical purposes: it will not be used in the final implementation, because a good term generator makes parsing superfluous at this task.

should denote the set of syntactically correct bit sequences (semantically, they may either terminate or diverge), i.e. the domain of the decoding function, i.e. the range of the coding function. Thus,

We can avoid referring to any code notion, if we modularize out function

and give it a separate name, e.g.

and notice that it can be defined directly in terms of CL-terms (we need not use any decoding concept any longer):

Thus, we transfer (lift) the notion of “length” from bit sequences to combinatory logic terms in an appropriate way. Let us call it, e.g. the “norm” of the term.

Thus, Chaitin's construction is grasped also as

Thus, we have no notions of “bit sequence”,“code”, “coding”, “decoding” at all. But their ghosts still haunt us: the definition of norm function looks rather strange without thinking on the fact that is was transferred from a concept of coding.

3.1 More natural norm functions (from CL terms)

Question:
If we already move away from the approaches referring to any code concept, then
could we define norm in other ways? E.g.

And is it worth doing it at all? The former one, at leat, had a good theoretical foundation (based on analysis, arithmetic and probability theory). This latter one is not so cleaner, that we should prefer it, so, lacking theoretical grounds.

What I really want is to exclude conceptually the notion of coding, and with it the notion of “syntactically incorrect versus syntactically correct but diverging”. Thus, taking into account only syntactically correct things, seeing only the choice of terminating versus non-terminating. Thus taking only termination vs nontermination into account, when calculating Chaitin's construction.

What I want to preserve:

it can be interpreted as a probability

it is a normal number, as if its digits (in binary representation) were generated by tossing a coin

thus I do not want to spoil these features.

3.1.1 Table for simpler CL-terms

Let us not take into account coding and thus excluding the notion of “syntactically incorrect coding” even conceptually.
Can we guess a good norm?

Binary tree pattern

Maximal depth, vertices, edges

Leafs, branches

So many CL-terms = how to count it

Terminating, ratio

So many till now, ratio till now

0, 1, 0

1, 0

2 = 2

2, 1

2, 1

1, 3, 2

2, 1

4, 1

6, 1

2, 5, 4

3, 2

8, 1

14, 1

2, 5, 4

3, 2

8, 1

22, 1

2, 7, 6

4, 3

16, 1

38, 1

4 Implementation

To do:
Writing a program in Haskell -- or in combinatory logic:-) -- which could help in making conjectures on combinatory logic-based Chaitin's constructions. It would make only approximations, in a similar way that most Mandelbrot plotting softwares work. The analogy:

they ask for a maximum limit of iterations, so that they can make a conjecture on convergence of a series;

this program will ask for the maximum limit of reducton steps, so that it can make a conjecture on termination (having-normal-form) of a CL term.

Explanation for this: non-termination of each actually examined CL-term cannot be proven by the program, but a good conjecture can be made: if termination does not take place in the given limit of reduction steps, then the actually examined CL-term is regarded as non-terminating.

4.1 Architecture

A CL term generator generates CL terms in “ascending order” (in terms of a theoretically appropriate “norm”), and by computing the norm of each CL-term, it approximates Chaitin's construction (at a given number of digits, and according to the given maximal limit of reduction steps).